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Question Number 128998 by bramlexs22 last updated on 12/Jan/21

Given sin x +cos x = (5/6) . Find the value of (1/(sin x)) + (1/(cos x)) .

$$\:\mathrm{Given}\:\mathrm{sin}\:\mathrm{x}\:+\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{5}}{\mathrm{6}}\:.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}}\:. \\ $$

Answered by liberty last updated on 12/Jan/21

From condition sin x + cos x = (5/6) we get 1+2sin x cos x = ((25)/(36)) 2sin x cos x = −((11)/(36)) or sin x cos x= −((11)/(72)) so (1/(sin x)) + (1/(cos x)) = ((sin x+cos x)/(sin x cos x)) = (5/6)×(−((72)/(11))) = −((60)/(11))

$$\mathrm{From}\:\mathrm{condition}\:\mathrm{sin}\:\mathrm{x}\:+\:\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{5}}{\mathrm{6}} \\ $$$$\:\mathrm{we}\:\mathrm{get}\:\mathrm{1}+\mathrm{2sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}\:=\:\frac{\mathrm{25}}{\mathrm{36}} \\ $$$$\:\mathrm{2sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}\:=\:−\frac{\mathrm{11}}{\mathrm{36}}\:\mathrm{or}\:\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}=\:−\frac{\mathrm{11}}{\mathrm{72}} \\ $$$$\mathrm{so}\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{5}}{\mathrm{6}}×\left(−\frac{\mathrm{72}}{\mathrm{11}}\right) \\ $$$$\:=\:−\frac{\mathrm{60}}{\mathrm{11}} \\ $$

Answered by MJS_new last updated on 12/Jan/21

s+c=(5/6) (s+c)^2 =((25)/(36)) ⇒ s^2 +c^2 +2sc=((25)/(36)) ⇒ sc=−((11)/(72)) (1/s)+(1/c)=((s+c)/(sc))=((5/6)/(−((11)/(72))))=−((60)/(11))

$${s}+{c}=\frac{\mathrm{5}}{\mathrm{6}} \\ $$$$\left({s}+{c}\right)^{\mathrm{2}} =\frac{\mathrm{25}}{\mathrm{36}}\:\Rightarrow\:{s}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{sc}=\frac{\mathrm{25}}{\mathrm{36}}\:\Rightarrow\:{sc}=−\frac{\mathrm{11}}{\mathrm{72}} \\ $$$$\frac{\mathrm{1}}{{s}}+\frac{\mathrm{1}}{{c}}=\frac{{s}+{c}}{{sc}}=\frac{\frac{\mathrm{5}}{\mathrm{6}}}{−\frac{\mathrm{11}}{\mathrm{72}}}=−\frac{\mathrm{60}}{\mathrm{11}} \\ $$

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